Algebra is abstract mathematics - let us make no bones about it - yet it is also applied mathematics in its best and purest form. It is not abstraction for its own sake, but abstraction for the sake of efficiency, power and insight. Algebra emerged from the struggle to solve concrete, physical problems in geometry, and succeeded after 2000 years of failure by other forms of mathematics. It did this by exposing the mathematical structure of geometry, and by providing the tools to analyse it. This is typical of the way algebra is applied; it is the best and purest form of application because it reveals the simplest and most universal mathematical structures. The present book aims to foster a proper appreciation of algebra by showing abstraction at work on concrete problems, the classical problems of construction by straightedge and compass. These problems originated in the time of Euclid, when geometry and number theory were paramount, and were not solved until th the 19 century, with the advent of abstract algebra. As we now know, alge bra brings about a unification of geometry, number theory and indeed most branches of mathematics. This is not really surprising when one has a historical understanding of the subject, which I also hope to impart.
Author(s): John Stillwell
Series: Undergraduate Texts in Mathematics
Edition: 1st
Publisher: Springer
Year: 2001
Language: English
Pages: 184
Tags: Geometry & Topology;Algebraic Geometry;Analytic Geometry;Differential Geometry;Non-Euclidean Geometries;Topology;Mathematics;Science & Math;Algebra;Abstract;Elementary;Intermediate;Linear;Pure Mathematics;Mathematics;Science & Math;Number Theory;Pure Mathematics;Mathematics;Science & Math;Algebra & Trigonometry;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique;Geometry;Mathematics;Science & Mathematics;New, Used & Rental Textbooks;Specialty Boutique
1.2 Straightedge and Compass Constructions......Page 1
1.3 The Constructible Numbers......Page 5
1.4 Some Famous Constructible Figures......Page 7
1.5 The Classical Construction Problems......Page 10
1.6 Quadratic and Cubic Equations......Page 11
1.7 Quartic Equations......Page 13
1.8 Solution by Radicals......Page 14
1.9 Discussion......Page 15
2.1 Natural Numbers......Page 18
2.2 Integers and Rational Numbers......Page 20
2.3 Divisibility......Page 22
2.4 The Euclidean Algorithm......Page 23
2.5 Unique Prime Factorisation......Page 25
2.6 Congruences......Page 26
2.7 Rings and Fields of Congruence Classes......Page 28
2.8 The Theorems of Fermat and Euler......Page 29
2.9 Fractions and the Euler Phi Function......Page 32
2.10 Discussion......Page 34
3.1 Irrational Numbers......Page 38
3.2 Existence and Meaning of Irrational Numbers......Page 40
3.3 The Real Numbers......Page 41
3.4 Arithmetic and Rational Functions on R......Page 42
3.5 Continuity and Completeness......Page 44
3.6 Complex Numbers......Page 46
3.7 Regular Polygons......Page 48
3.8 The Fundamental Theorem of Algebra......Page 51
3.9 Discussion......Page 53
4.1 Polynomials over a Field......Page 57
4.2 Divisibility......Page 58
4.3 Unique Factorisation......Page 61
4.4 Congruences......Page 62
4.5 The Fields F(\alpha)......Page 63
4.6 Gauss's Lemma......Page 65
4.7 Eisenstein's Irreducibility Criterion......Page 67
4.8 Cyclotomic Polynomials......Page 69
4.9 Irreducibility of Cyclotomic Polynomials......Page 71
4.10 Discussion......Page 73
5.2 Algebraic Numbers and Fields......Page 76
5.3 Algebraic Elements over an Arbitrary Field......Page 77
5.4 Degree of a Field over a Subfield......Page 78
5.5 Degree of an Iterated Extension......Page 81
5.6 Degree of Constructible Numbers......Page 83
5.7 Regular n-gons......Page 85
5.8 Discussion......Page 86
6.1 Ring and Field Isomorphisms......Page 89
6.2 Isomorphisms of Q(\alpha) and F(\alpha)......Page 91
6.3 Extending Fields and Isomorphisms......Page 94
6.4 Automorphisms and Groups......Page 97
6.5 Function Fields and Symmetric Functions......Page 98
6.6 Cyclotomic Fields......Page 100
6.7 The Chinese Remainder Theorem......Page 101
6.8 Homomorphisms and Quotient Rings......Page 103
6.9 Discussion......Page 105
7.1 Why Groups?......Page 108
7.2 Cayley's Theorem......Page 109
7.3 Abelian Groups......Page 111
7.4 Dihedral Groups......Page 112
7.5 Permutation Groups......Page 115
7.6 Permutation Groups in Geometry......Page 116
7.7 Subgroups and Cosets......Page 119
7.8 Normal Subgroups......Page 121
7.9 Homomorphisms......Page 122
7.10 Discussion......Page 125
8.1 Galois Groups......Page 128
8.2 Solution by Radicals......Page 130
8.3 Structure of Radical Extensions......Page 132
8.4 Nonexistence of Solutions by Radicals when n \geq 5......Page 134
8.5 Quintics with Integer Coefficients......Page 136
8.6 Unsolvable Quintic Equations with Integer Coefficients......Page 138
8.7 Primitive Roots......Page 139
8.8 Finite Abelian Groups......Page 141
8.9 Discussion......Page 143
9.1 The Theorem of the Primitive Element......Page 146
9.2 Conjugate Fields and Splitting Fields......Page 148
9.3 Fixed Fields......Page 150
9.4 Conjugate Intermediate Fields......Page 152
9.5 Normal Extensions with Solvable Galois Group......Page 154
9.6 Cyclic Extensions......Page 155
9.7 Construction of the Radical Extension......Page 156
9.8 Construction of Regular p-gons......Page 157
9.9 Division of Arbitrary Angles......Page 159
9.10 Discussion......Page 160
References......Page 162
Index......Page 170
